An Optimal Algorithm for Prufer Codes *
|
|
- Briana Bradford
- 6 years ago
- Views:
Transcription
1 J. Software Engneerng & Applcatons, 2009, 2: do: /jsea Publshed Onlne July 2009 ( An Optmal Algorthm for Prufer Codes * Xaodong Wang 1, 2, Le Wang 3, Yngje Wu 1 1 Department of Computer Scence, Fuzhou Unversty, Fuzhou, Chna; 2 Department of Computer Scence, Quanzhou Normal Unversty, Quanzhou, Chna; 3 College of Computng, Georga Insttute of Technology, Atlanta GA 30332, USA. Emal: wangxd@fzu.edu.cn Receved February 19 th, 2009; revsed February 23 rd, 2009; accepted February 24 th, ABSTRACT Ths paper studes the algorthms for codng and decodng Prufer codes of a labeled tree. The algorthms for codng and decodng Prufer codes of a labeled tree n the lteratures requre Onlogn) ( tme usually. Although there exst lnear tme algorthms for Prufer-lke codes [1,2,3], the algorthms utlze the nteger sortng algorthms. The specal range of the ntegers to be sorted s utlzed to obtan a lnear tme nteger sortng algorthm. The Prufer code problem s reduced to nteger sortng. In ths paper we consder the Prufer code problem n a dfferent angle and a more drect manner. We start from a naïve algorthm, then mproved t gradually and fnally we obtan a very practcal lnear tme algorthm. The technques we used n ths paper are of nterest n ther own rght. Keywords: Desgn of Algorthm, Labeled Trees, Prufer Codes, Integer Sortng 1. Introducton * Supported by Natural Scence Foundaton of Chna under Grant No and Natural Scence Foundaton of Fujan under Grant No. A Labeled trees are of nterest n practcal and theoretcal areas of computer scence. For example, Ethernet has a unque path between termnal devces, thus beng a tree: labelng the tree nodes s necessary to unquely dentfy each devce n the network. An nterestng alternatve to the usual representatons of tree data structures n computer memores s based on codng labeled trees by means of strngs of node labels. Ths representaton was frst used n the proof of Cayley s theorem [4] to show a one-to-one correspondence between free labeled trees on n nodes and strngs of length n-2. In addton to ths purely mathematcal use, strng-based codng of trees has many practcal applcatons. For nstance, they make t possble to generate random unformly dstrbuted trees and random connected graphs: the generaton of a random strng followed by the use of a fast decodng algorthm s typcally more effcent than random tree generaton by the addton of edges, snce n the latter case one must pay attenton not to ntroduce cycles. In addton, tree codes are employed n genetc algorthms, where chromosomes n the populaton are represented as strngs of ntegers, and n heurstcs for computng mnmum spannng trees wth addtonal constrants, e.g., on the number of leaves or on the dameter of the tree tself. Not last, tree codes are used for data compresson and for computng the tree and forest volumes of graphs. Let T be a labeled tree whose nodes are numbered from 0 to n-1. For some vertex v n T, the degree of v, denoted by d[v], s the number of edges ncdent to v. If d[v]=1, then v s called a leaf. Accordng to Prufer s proof, any sequence of n-2 numbers, each number n {0,1,,n-1} can determne a unque labeled tree of n nodes. Such a number sequence s the Prufer code of a labeled tree. Algorthm A s the straghtforward mplementaton of Prufer s proof. Algorthm A Input: A labeled tree T of n nodes as a lst of n-1 edges. Output: c, the Prufer code of T. Method: Step 1. B {0,1,,n-1}. Step 2. For 0 to n-3 do Step 2.1. x mn{k B: k s a leaf }. Step 2.2. B B-{x}. Step 2.3. Remove x and ts ncdent edge (x, y) from T. Step 2.4. c[] y. Step 3. Return c. End of Algorthm For example, let the nput labeled tree T be the graph depcted n Fgure 1. After the Algorthm A termnates, c = [2,4,0,1,3,3] whch s the Prufer code of T. Copyrght 2009 ScRes
2 112 An Optmal Algorthm for Prufer Codes Fgure 1. A labelled tree The algorthms for codng and decodng Prufer codes of a labeled tree n the lteratures requre Onlogn) ( tme usually. As stated n [3], although the problem of producng a Prufer code n lnear tme s an exercse n two books [5, 6], there exsts no explct publcaton of a soluton. In [3] an On ( ) tme algorthm for generatng a Prufer code from a labeled tree was descrbed, but the On ( ) tme algorthm utlzed the nteger sortng algorthms. The specal range of the ntegers to be sorted was utlzed to obtan a lnear tme nteger sortng algorthm. The Prufer code problem s reduced to nteger sortng. In ths paper we consder the Prufer code problem n a dfferent angle and present a very practcal lnear tme algorthm drectly. The technques we used n ths paper are of nterest n ther own rght. 2. A Lnear Tme Algorthm for Codng The most tme consumng step n Algorthm A s Step 2. The leaf elmnaton scheme of the Prufer code mplctly defnes a root for the labeled tree T. Actually, t s easy to see that the last element n the code s n-1 whch s the root of the labeled tree T. Gven a lst of n-1 edges, we can buld the parent array f and the degree array d of the labeled tree T wth only On ( ) preprocessng tme, whch allows checkng and updatng a node's degree n O(1) tme. Wth ths preprocessng, the Step 2.3 of Algorthm A can be mplemented n O(1) tme. Usng a heap, the leaf wth the smallest number can be found n Onl ( ogn) tme. Hence, Step 2 takes Onlogn ( ) tme totally. The tme complexty of Algorthm A s also Onl ( ogn). We can mprove the algorthms further. An nsght nto the problem s that for the heap operatons n the Step 2.1 of Algorthm A, there s a specal knd of nodes that they are deleted from the heap mmedately after they are nserted nto the heap. Ths knd of nodes can be treated easly wthout heap operaton. The remaned heap operaton can be replaced by a lnear scan of the degree array d. Therefore the total tme to fnd x n the Step 2.1 s reduced to O(n). The lnear tme algorthm can be presented detaled as follows. Codng Algorthm: 1: ndex x mn{0 k<n: d[k]=1} 2: for 0 to n-3 do 3: y f[x] 4: c[] y 5: d[y] d[y]-1 6: f y<ndex and d[y]=1 then x y 7: else ndex x mn{ndex<k<n: d[k]=1} In the algorthm descrbed above, d[v] s the degree of node v and f[v] s the parent node of the node v. The varable ndex s a cursor of degree array d. On lne 6, when node y becomes a leaf and the label of y s less than the current cursor ndex, the node y must be the knd of nodes that are deleted from the heap mmedately after they are nserted nto the heap. In ths case, the node y becomes the next node wth mnmal label. Otherwse, on lne 7 the cursor ndex moves to the next leaf node n the degree array d. The computng tme of lne 1 and lne 7 s On ( ), snce the cursor ndex goes through the degree array d from left to rght once. The remanng tme s clearly On) (, leadng to On ( ) complexty for the entre algorthm. 3. A Lnear Tme Algorthm for Decodng Decodng s to buld the tree T correspondng to the gven Prufer code c. As far as c s computed, each node label n t represents the parent of a leaf elmnated from T. Hence, n order to reconstruct T, t s suffcent to compute the ordered sequence of labels of the elmnated leaves, say s: for each {0,1,,n-1}, the par (c[], s[]) wll thus be an edge n the tree. We frst observe that the leaves of T are exactly those nodes that do not appear n the code, as they are not parents of any node. Each nternal node, say v, n general may appear n c more than once; each appearance corresponds to the elmnaton of one of ts chldren, and therefore to decreasng the degree of v by 1. After the rghtmost occurrence n the code, v s clearly a leaf and thus becomes a canddate for beng elmnated. Therefore, the tmes of a node v appears n c s exactly the degree of v mnus 1. A lnear scan of code array c can determne the degree array d. Usng a heap, the leaf wth the smallest number can be found n O( nlogn ) tme, leadng an Onlogn ( ) tme decodng algorthm. Lke the codng algorthm, we can mprove the algorthms further. An nsght nto the problem s also that for the heap operatons of the decodng algorthm, there s a specal knd of nodes that they are deleted from the heap mmedately after they are nserted nto the heap. Ths knd of nodes can be treated easly wthout heap operaton. The remaned heap operaton can be replaced by a lnear scan of the degree array d. Copyrght 2009 ScRes
3 An Optmal Algorthm for Prufer Codes 113 The lnear tme algorthm can be presented detaled as follows. Decodng Algorthm: 1: ndex x mn{0 k<n: d[k]=1} 2: for 0 to n-2 do 3: y c[] 4: add edge (x,y) to T 5: d[y] d[y]-1 6: f y<ndex and d[y]=1 then x y 7: else ndex x mn{ndex<k<n: d[k]=1} Lke the codng algorthm, the tme requred by the decodng s also On ( ). 4. Examples We use the labeled tree T depcted n Fgure 1 as an example to demonstrate the algorthms for codng and decodng Prufer codes descrbed above. The nput of the labeled tree T s an edge lst {0,1}{0,4}{1,3}{4,2}{3,6} {3,7}{2,5}. There are 8(n) nodes labeled 0,1,2,3,4,5,6,7 and 7(n-1) edges. It s easy to see that the last element 7 s the root of the labeled tree T. For the gven edge lst, we can buld the parent array f and the degree array d of the labeled tree T by a depth frst search wth only On ( ) tme. Wth ths two arrays, the Step 2.3 of Algorthm A can be mplemented n O(1) tme. For our lnear tme codng algorthm, we frst go through the degree array d to fnd an ndex such that ndex=mn{0 k<n: d[k]=1}. It s clear that the ndex equals 5 for the frst tme n our example as shown n Table 1. Table 1. The parent array f and the degree array d of T f[] d[] ndex Table 2. After edge {2,5} s deleted [5] 6 7 f[] d[] ndex Table 3. After edge {4,2} s deleted 0 1 [2] 3 4 [5] 6 7 f[] d[] ndex Table 4. After the edges {0,4}, {0,1} and {1,3} are deleted [0] [1] [2] 3 [4] [5] 6 7 f[] d[] ndex Table 5. After the edge {3,6} s deleted [0] [1] [2] 3 [4] [5] [6] 7 f[] d[] ndex f[ndex]=f[5]=2 s the frst value of the Prufer code c. The edge {2,5} s deleted and d[2] s decreased by 1 on lne 5 of the codng algorthm. The status of the tree T now becomes Table 2. Follows the codng algorthm on lne 6 the next node to be deleted s 2. The father node of 2 s node 4, the next value of the Prufer code c. The edge {4,2} s deleted and d[4] s decreased by 1 on lne 5 of the codng algorthm. The status of the tree T now becomes Table 3. Smlarly, n the next three steps we obtan the next three values 0,1 and 3 of the Prufer code c. The edges {0,4}, {0,1} and {1,3} are deleted accordngly. The Prufer code we have obtaned up to now s (2,4,0,1,3). The status of the tree T now becomes Table 4. Look at the Table 4, the next node to be deleted s 6 whch s determned on lne 7 of our codng algorthm. We do not scan the degree array d from the begnnng. We scan the degree array d from ndex (5) to rght and the ndex s moved to 6. Ths s a key pont to see the algorthm runnng n lnear tme. The father node of 6 s node 3, the next value of the Prufer code c. The edge {3,6} s deleted and d[3] s decreased by 1 on lne 5 of the codng algorthm. The status of the tree T now becomes Table 5. Copyrght 2009 ScRes
4 114 An Optmal Algorthm for Prufer Codes Table 6. The nto status of the decodng algorthm c[] d[] ndex Table 7. After edge {2,5} s added [5] 6 7 c[] d[] ndex Table 8. After edge {2,4} s added 0 1 [2] 3 4 [5] 6 7 c[] d[] ndex We now have the n-2 values of the Prufer code (2,4,0,1,3,3). The algorthm termnates. Usng the same labeled tree T depcted n Fgure 1 as an example we can demonstrate the algorthm for decodng Prufer codes as follows. The nput of the decodng algorthm s the Prufer code c of the tree T. In our example the Prufer code of the tree T s (2,4,0,1,3,3). We fst noted that the tmes of a node v appears n the Prufer code c s exactly the degree of v mnus 1. A lnear scan of code array c can determne the degree array d as shown n Table 6. For our lnear tme decodng algorthm, we frst go through the degree array d to fnd an ndex such that ndex=mn{0 k<n:d[k]=1}. It s clear that the ndex equals 5 for the frst tme n our example as shown n Table 6 whch s the frst leaf node deleted n the codng algorthm. c[0]=2 s the other node label of the edge to be added. The edge {2,5} s added and d[2] s decreased by 1 on lne 5 of the decodng algorthm. The status of the tree T now becomes Table 7. Follows the decodng algorthm on lne 6 the next node to be added s 2. c[1]=4 s the other node label of the edge to be added. The edge {2,4} s added and d[4] s decreased by 1 on lne 5 of the decodng algorthm. The status of the tree T now becomes Table 8. Smlarly, n the next three steps we obtan the next three edges {0,4}, {0,1} and {1,3}. The status of the tree T now becomes Table 9. Look at the Table 9. The next node to be added s 6 whch s determned on lne 7 of our decodng algorthm. We do not scan the degree array d from the begnnng. We scan the degree array d from ndex (5) to rght and the ndex s moved to 6. Ths s a key pont to see the decodng algorthm runnng n lnear tme. The other node label of the edge to be added s c[5]=3. The edge {3,6} s added and d[3] s decreased by 1 on lne 5 of the decodng algorthm. The status of the tree T now becomes Table 10. Follows the decodng algorthm on lne 6 the next node to be added s 3. c[6]=7 s the other node label of the edge to be added. The edge {3,7} s added. We now have the n-1 edges of the tree T. The decodng algorthm termnates. 5. Conclusons Prufer codes for labeled trees have many practcal applcatons. The algorthms for codng and decodng Prufer codes of a labeled tree are of nterest n practcal and theoretcal areas of computer scence. The exstng lnear tme algorthms for Prufer-lke codes utlzed the nteger sortng algorthms. The specal range of the ntegers to be sorted s utlzed to obtan a lnear tme nteger sortng algorthm. The optmal algorthms for codng and decodng Prufer codes presented n ths paper are very practcal lnear tme algorthms. The technques used n these algorthms are of nterest n ther own rght. Table 9. After the edges {0,4}, {0,1} and {1,3} are added [0] [1] [2] 3 [4] [5] 6 7 c[] d[] ndex Table 10. After the edge {3,6} s added [0] [1] [2] 3 [4] [5] [6] 7 c[] d[] ndex Copyrght 2009 ScRes
5 An Optmal Algorthm for Prufer Codes Acknowledgments Ths work was supported by Natural Scence Foundaton of Chna under Grant No and Natural Scence Foundaton of Fujan under Grant No. A REFERENCES [1] S. Camnt, I. Fnocch, and R. Petresch, A unfed approach to codng labeled trees, n Proceedngs of the 6th Latn Amercan Symposum on Theoretcal Informatcs (LATIN 04), LNCS 2976, pp , [2] S. Camnt, I. Fnocch, and R. Petresch, On codng labeled trees, To appear on Theoretcal Computer Scence, [3] H. C. Chen and Y. L. Wang, An effcent algorthm for generatng Prufer codes from labeled trees, Theory of Computng Systems, Vol. 33, pp , [4] A. Cayley, A theorem on trees, Quarterly Journal of Mathematcs, Vol. 23, pp , [5] L. Devroye, Non-unform random varate generaton, Sprnger-Verlag, New York, Exercse 2, pp. 666, [6] A. Njenhus and H. S. Wlf, Combnatoral algorthms for computers and calculators, Second Edton, Academc Press, New York, Exercse 46, pp. 293, Copyrght 2009 ScRes
Parallelism for Nested Loops with Non-uniform and Flow Dependences
Parallelsm for Nested Loops wth Non-unform and Flow Dependences Sam-Jn Jeong Dept. of Informaton & Communcaton Engneerng, Cheonan Unversty, 5, Anseo-dong, Cheonan, Chungnam, 330-80, Korea. seong@cheonan.ac.kr
More informationRelated-Mode Attacks on CTR Encryption Mode
Internatonal Journal of Network Securty, Vol.4, No.3, PP.282 287, May 2007 282 Related-Mode Attacks on CTR Encrypton Mode Dayn Wang, Dongda Ln, and Wenlng Wu (Correspondng author: Dayn Wang) Key Laboratory
More informationNon-Split Restrained Dominating Set of an Interval Graph Using an Algorithm
Internatonal Journal of Advancements n Research & Technology, Volume, Issue, July- ISS - on-splt Restraned Domnatng Set of an Interval Graph Usng an Algorthm ABSTRACT Dr.A.Sudhakaraah *, E. Gnana Deepka,
More informationCompiler Design. Spring Register Allocation. Sample Exercises and Solutions. Prof. Pedro C. Diniz
Compler Desgn Sprng 2014 Regster Allocaton Sample Exercses and Solutons Prof. Pedro C. Dnz USC / Informaton Scences Insttute 4676 Admralty Way, Sute 1001 Marna del Rey, Calforna 90292 pedro@s.edu Regster
More informationLoad Balancing for Hex-Cell Interconnection Network
Int. J. Communcatons, Network and System Scences,,, - Publshed Onlne Aprl n ScRes. http://www.scrp.org/journal/jcns http://dx.do.org/./jcns.. Load Balancng for Hex-Cell Interconnecton Network Saher Manaseer,
More informationSolving two-person zero-sum game by Matlab
Appled Mechancs and Materals Onlne: 2011-02-02 ISSN: 1662-7482, Vols. 50-51, pp 262-265 do:10.4028/www.scentfc.net/amm.50-51.262 2011 Trans Tech Publcatons, Swtzerland Solvng two-person zero-sum game by
More informationProblem Set 3 Solutions
Introducton to Algorthms October 4, 2002 Massachusetts Insttute of Technology 6046J/18410J Professors Erk Demane and Shaf Goldwasser Handout 14 Problem Set 3 Solutons (Exercses were not to be turned n,
More informationSequential search. Building Java Programs Chapter 13. Sequential search. Sequential search
Sequental search Buldng Java Programs Chapter 13 Searchng and Sortng sequental search: Locates a target value n an array/lst by examnng each element from start to fnsh. How many elements wll t need to
More informationSorting Review. Sorting. Comparison Sorting. CSE 680 Prof. Roger Crawfis. Assumptions
Sortng Revew Introducton to Algorthms Qucksort CSE 680 Prof. Roger Crawfs Inserton Sort T(n) = Θ(n 2 ) In-place Merge Sort T(n) = Θ(n lg(n)) Not n-place Selecton Sort (from homework) T(n) = Θ(n 2 ) In-place
More informationAn Application of the Dulmage-Mendelsohn Decomposition to Sparse Null Space Bases of Full Row Rank Matrices
Internatonal Mathematcal Forum, Vol 7, 2012, no 52, 2549-2554 An Applcaton of the Dulmage-Mendelsohn Decomposton to Sparse Null Space Bases of Full Row Rank Matrces Mostafa Khorramzadeh Department of Mathematcal
More informationDesign and Analysis of Algorithms
Desgn and Analyss of Algorthms Heaps and Heapsort Reference: CLRS Chapter 6 Topcs: Heaps Heapsort Prorty queue Huo Hongwe Recap and overvew The story so far... Inserton sort runnng tme of Θ(n 2 ); sorts
More informationCourse Introduction. Algorithm 8/31/2017. COSC 320 Advanced Data Structures and Algorithms. COSC 320 Advanced Data Structures and Algorithms
Course Introducton Course Topcs Exams, abs, Proects A quc loo at a few algorthms 1 Advanced Data Structures and Algorthms Descrpton: We are gong to dscuss algorthm complexty analyss, algorthm desgn technques
More informationOn Some Entertaining Applications of the Concept of Set in Computer Science Course
On Some Entertanng Applcatons of the Concept of Set n Computer Scence Course Krasmr Yordzhev *, Hrstna Kostadnova ** * Assocate Professor Krasmr Yordzhev, Ph.D., Faculty of Mathematcs and Natural Scences,
More information6.854 Advanced Algorithms Petar Maymounkov Problem Set 11 (November 23, 2005) With: Benjamin Rossman, Oren Weimann, and Pouya Kheradpour
6.854 Advanced Algorthms Petar Maymounkov Problem Set 11 (November 23, 2005) Wth: Benjamn Rossman, Oren Wemann, and Pouya Kheradpour Problem 1. We reduce vertex cover to MAX-SAT wth weghts, such that the
More informationON SOME ENTERTAINING APPLICATIONS OF THE CONCEPT OF SET IN COMPUTER SCIENCE COURSE
Yordzhev K., Kostadnova H. Інформаційні технології в освіті ON SOME ENTERTAINING APPLICATIONS OF THE CONCEPT OF SET IN COMPUTER SCIENCE COURSE Yordzhev K., Kostadnova H. Some aspects of programmng educaton
More informationFast Computation of Shortest Path for Visiting Segments in the Plane
Send Orders for Reprnts to reprnts@benthamscence.ae 4 The Open Cybernetcs & Systemcs Journal, 04, 8, 4-9 Open Access Fast Computaton of Shortest Path for Vstng Segments n the Plane Ljuan Wang,, Bo Jang
More informationFor instance, ; the five basic number-sets are increasingly more n A B & B A A = B (1)
Secton 1.2 Subsets and the Boolean operatons on sets If every element of the set A s an element of the set B, we say that A s a subset of B, or that A s contaned n B, or that B contans A, and we wrte A
More informationThe Greedy Method. Outline and Reading. Change Money Problem. Greedy Algorithms. Applications of the Greedy Strategy. The Greedy Method Technique
//00 :0 AM Outlne and Readng The Greedy Method The Greedy Method Technque (secton.) Fractonal Knapsack Problem (secton..) Task Schedulng (secton..) Mnmum Spannng Trees (secton.) Change Money Problem Greedy
More informationThe Shortest Path of Touring Lines given in the Plane
Send Orders for Reprnts to reprnts@benthamscence.ae 262 The Open Cybernetcs & Systemcs Journal, 2015, 9, 262-267 The Shortest Path of Tourng Lnes gven n the Plane Open Access Ljuan Wang 1,2, Dandan He
More informationF Geometric Mean Graphs
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 2 (December 2015), pp. 937-952 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) F Geometrc Mean Graphs A.
More informationCMPS 10 Introduction to Computer Science Lecture Notes
CPS 0 Introducton to Computer Scence Lecture Notes Chapter : Algorthm Desgn How should we present algorthms? Natural languages lke Englsh, Spansh, or French whch are rch n nterpretaton and meanng are not
More informationNUMERICAL SOLVING OPTIMAL CONTROL PROBLEMS BY THE METHOD OF VARIATIONS
ARPN Journal of Engneerng and Appled Scences 006-017 Asan Research Publshng Network (ARPN). All rghts reserved. NUMERICAL SOLVING OPTIMAL CONTROL PROBLEMS BY THE METHOD OF VARIATIONS Igor Grgoryev, Svetlana
More informationCHAPTER 2 DECOMPOSITION OF GRAPHS
CHAPTER DECOMPOSITION OF GRAPHS. INTRODUCTION A graph H s called a Supersubdvson of a graph G f H s obtaned from G by replacng every edge uv of G by a bpartte graph,m (m may vary for each edge by dentfyng
More informationModule Management Tool in Software Development Organizations
Journal of Computer Scence (5): 8-, 7 ISSN 59-66 7 Scence Publcatons Management Tool n Software Development Organzatons Ahmad A. Al-Rababah and Mohammad A. Al-Rababah Faculty of IT, Al-Ahlyyah Amman Unversty,
More informationCS1100 Introduction to Programming
Factoral (n) Recursve Program fact(n) = n*fact(n-) CS00 Introducton to Programmng Recurson and Sortng Madhu Mutyam Department of Computer Scence and Engneerng Indan Insttute of Technology Madras nt fact
More informationGSLM Operations Research II Fall 13/14
GSLM 58 Operatons Research II Fall /4 6. Separable Programmng Consder a general NLP mn f(x) s.t. g j (x) b j j =. m. Defnton 6.. The NLP s a separable program f ts objectve functon and all constrants are
More informationAn Iterative Solution Approach to Process Plant Layout using Mixed Integer Optimisation
17 th European Symposum on Computer Aded Process Engneerng ESCAPE17 V. Plesu and P.S. Agach (Edtors) 2007 Elsever B.V. All rghts reserved. 1 An Iteratve Soluton Approach to Process Plant Layout usng Mxed
More informationCluster Analysis of Electrical Behavior
Journal of Computer and Communcatons, 205, 3, 88-93 Publshed Onlne May 205 n ScRes. http://www.scrp.org/ournal/cc http://dx.do.org/0.4236/cc.205.350 Cluster Analyss of Electrcal Behavor Ln Lu Ln Lu, School
More informationA Facet Generation Procedure. for solving 0/1 integer programs
A Facet Generaton Procedure for solvng 0/ nteger programs by Gyana R. Parja IBM Corporaton, Poughkeepse, NY 260 Radu Gaddov Emery Worldwde Arlnes, Vandala, Oho 45377 and Wlbert E. Wlhelm Teas A&M Unversty,
More informationGreedy Technique - Definition
Greedy Technque Greedy Technque - Defnton The greedy method s a general algorthm desgn paradgm, bult on the follong elements: confguratons: dfferent choces, collectons, or values to fnd objectve functon:
More informationCE 221 Data Structures and Algorithms
CE 1 ata Structures and Algorthms Chapter 4: Trees BST Text: Read Wess, 4.3 Izmr Unversty of Economcs 1 The Search Tree AT Bnary Search Trees An mportant applcaton of bnary trees s n searchng. Let us assume
More informationConstructing Minimum Connected Dominating Set: Algorithmic approach
Constructng Mnmum Connected Domnatng Set: Algorthmc approach G.N. Puroht and Usha Sharma Centre for Mathematcal Scences, Banasthal Unversty, Rajasthan 304022 usha.sharma94@yahoo.com Abstract: Connected
More informationProblem Definitions and Evaluation Criteria for Computational Expensive Optimization
Problem efntons and Evaluaton Crtera for Computatonal Expensve Optmzaton B. Lu 1, Q. Chen and Q. Zhang 3, J. J. Lang 4, P. N. Suganthan, B. Y. Qu 6 1 epartment of Computng, Glyndwr Unversty, UK Faclty
More informationInsertion Sort. Divide and Conquer Sorting. Divide and Conquer. Mergesort. Mergesort Example. Auxiliary Array
Inserton Sort Dvde and Conquer Sortng CSE 6 Data Structures Lecture 18 What f frst k elements of array are already sorted? 4, 7, 1, 5, 1, 16 We can shft the tal of the sorted elements lst down and then
More informationSum of Linear and Fractional Multiobjective Programming Problem under Fuzzy Rules Constraints
Australan Journal of Basc and Appled Scences, 2(4): 1204-1208, 2008 ISSN 1991-8178 Sum of Lnear and Fractonal Multobjectve Programmng Problem under Fuzzy Rules Constrants 1 2 Sanjay Jan and Kalash Lachhwan
More informationCSCI 104 Sorting Algorithms. Mark Redekopp David Kempe
CSCI 104 Sortng Algorthms Mark Redekopp Davd Kempe Algorthm Effcency SORTING 2 Sortng If we have an unordered lst, sequental search becomes our only choce If we wll perform a lot of searches t may be benefcal
More informationA mathematical programming approach to the analysis, design and scheduling of offshore oilfields
17 th European Symposum on Computer Aded Process Engneerng ESCAPE17 V. Plesu and P.S. Agach (Edtors) 2007 Elsever B.V. All rghts reserved. 1 A mathematcal programmng approach to the analyss, desgn and
More informationR s s f. m y s. SPH3UW Unit 7.3 Spherical Concave Mirrors Page 1 of 12. Notes
SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 1 of 1 Notes Physcs Tool box Concave Mrror If the reflectng surface takes place on the nner surface of the sphercal shape so that the centre of the mrror bulges
More informationLearning the Kernel Parameters in Kernel Minimum Distance Classifier
Learnng the Kernel Parameters n Kernel Mnmum Dstance Classfer Daoqang Zhang 1,, Songcan Chen and Zh-Hua Zhou 1* 1 Natonal Laboratory for Novel Software Technology Nanjng Unversty, Nanjng 193, Chna Department
More informationProgramming in Fortran 90 : 2017/2018
Programmng n Fortran 90 : 2017/2018 Programmng n Fortran 90 : 2017/2018 Exercse 1 : Evaluaton of functon dependng on nput Wrte a program who evaluate the functon f (x,y) for any two user specfed values
More informationDetermining the Optimal Bandwidth Based on Multi-criterion Fusion
Proceedngs of 01 4th Internatonal Conference on Machne Learnng and Computng IPCSIT vol. 5 (01) (01) IACSIT Press, Sngapore Determnng the Optmal Bandwdth Based on Mult-crteron Fuson Ha-L Lang 1+, Xan-Mn
More informationConcurrent Apriori Data Mining Algorithms
Concurrent Apror Data Mnng Algorthms Vassl Halatchev Department of Electrcal Engneerng and Computer Scence York Unversty, Toronto October 8, 2015 Outlne Why t s mportant Introducton to Assocaton Rule Mnng
More informationA New Approach For the Ranking of Fuzzy Sets With Different Heights
New pproach For the ankng of Fuzzy Sets Wth Dfferent Heghts Pushpnder Sngh School of Mathematcs Computer pplcatons Thapar Unversty, Patala-7 00 Inda pushpndersnl@gmalcom STCT ankng of fuzzy sets plays
More informationPrivate Information Retrieval (PIR)
2 Levente Buttyán Problem formulaton Alce wants to obtan nformaton from a database, but she does not want the database to learn whch nformaton she wanted e.g., Alce s an nvestor queryng a stock-market
More informationNGPM -- A NSGA-II Program in Matlab
Verson 1.4 LIN Song Aerospace Structural Dynamcs Research Laboratory College of Astronautcs, Northwestern Polytechncal Unversty, Chna Emal: lsssswc@163.com 2011-07-26 Contents Contents... 1. Introducton...
More informationA fast algorithm for color image segmentation
Unersty of Wollongong Research Onlne Faculty of Informatcs - Papers (Arche) Faculty of Engneerng and Informaton Scences 006 A fast algorthm for color mage segmentaton L. Dong Unersty of Wollongong, lju@uow.edu.au
More informationCSE 326: Data Structures Quicksort Comparison Sorting Bound
CSE 326: Data Structures Qucksort Comparson Sortng Bound Steve Setz Wnter 2009 Qucksort Qucksort uses a dvde and conquer strategy, but does not requre the O(N) extra space that MergeSort does. Here s the
More informationCHARUTAR VIDYA MANDAL S SEMCOM Vallabh Vidyanagar
CHARUTAR VIDYA MANDAL S SEMCOM Vallabh Vdyanagar Faculty Name: Am D. Trved Class: SYBCA Subject: US03CBCA03 (Advanced Data & Fle Structure) *UNIT 1 (ARRAYS AND TREES) **INTRODUCTION TO ARRAYS If we want
More informationThe Research of Ellipse Parameter Fitting Algorithm of Ultrasonic Imaging Logging in the Casing Hole
Appled Mathematcs, 04, 5, 37-3 Publshed Onlne May 04 n ScRes. http://www.scrp.org/journal/am http://dx.do.org/0.436/am.04.584 The Research of Ellpse Parameter Fttng Algorthm of Ultrasonc Imagng Loggng
More informationA Five-Point Subdivision Scheme with Two Parameters and a Four-Point Shape-Preserving Scheme
Mathematcal and Computatonal Applcatons Artcle A Fve-Pont Subdvson Scheme wth Two Parameters and a Four-Pont Shape-Preservng Scheme Jeqng Tan,2, Bo Wang, * and Jun Sh School of Mathematcs, Hefe Unversty
More informationCordial and 3-Equitable Labeling for Some Star Related Graphs
Internatonal Mathematcal Forum, 4, 009, no. 31, 1543-1553 Cordal and 3-Equtable Labelng for Some Star Related Graphs S. K. Vadya Department of Mathematcs, Saurashtra Unversty Rajkot - 360005, Gujarat,
More informationClassifier Selection Based on Data Complexity Measures *
Classfer Selecton Based on Data Complexty Measures * Edth Hernández-Reyes, J.A. Carrasco-Ochoa, and J.Fco. Martínez-Trndad Natonal Insttute for Astrophyscs, Optcs and Electroncs, Lus Enrque Erro No.1 Sta.
More informationINTEGER PROGRAMMING MODELING FOR THE CHINESE POSTMAN PROBLEMS
INTEGER PROGRAMMING MODELING FOR THE CHINESE POSTMAN PROBLEMS ABSTRACT Feng Junwen School of Economcs and Management, Nanng Unversty of Scence and Technology, Nanng, 2009, Chna As far as the tradtonal
More informationSLAM Summer School 2006 Practical 2: SLAM using Monocular Vision
SLAM Summer School 2006 Practcal 2: SLAM usng Monocular Vson Javer Cvera, Unversty of Zaragoza Andrew J. Davson, Imperal College London J.M.M Montel, Unversty of Zaragoza. josemar@unzar.es, jcvera@unzar.es,
More informationthe nber of vertces n the graph. spannng tree T beng part of a par of maxmally dstant trees s called extremal. Extremal trees are useful n the mxed an
On Central Spannng Trees of a Graph S. Bezrukov Unverstat-GH Paderborn FB Mathematk/Informatk Furstenallee 11 D{33102 Paderborn F. Kaderal, W. Poguntke FernUnverstat Hagen LG Kommunkatonssysteme Bergscher
More informationToday s Outline. Sorting: The Big Picture. Why Sort? Selection Sort: Idea. Insertion Sort: Idea. Sorting Chapter 7 in Weiss.
Today s Outlne Sortng Chapter 7 n Wess CSE 26 Data Structures Ruth Anderson Announcements Wrtten Homework #6 due Frday 2/26 at the begnnng of lecture Proect Code due Mon March 1 by 11pm Today s Topcs:
More informationMeta-heuristics for Multidimensional Knapsack Problems
2012 4th Internatonal Conference on Computer Research and Development IPCSIT vol.39 (2012) (2012) IACSIT Press, Sngapore Meta-heurstcs for Multdmensonal Knapsack Problems Zhbao Man + Computer Scence Department,
More informationBridges and cut-vertices of Intuitionistic Fuzzy Graph Structure
Internatonal Journal of Engneerng, Scence and Mathematcs (UGC Approved) Journal Homepage: http://www.jesm.co.n, Emal: jesmj@gmal.com Double-Blnd Peer Revewed Refereed Open Access Internatonal Journal -
More informationA Binarization Algorithm specialized on Document Images and Photos
A Bnarzaton Algorthm specalzed on Document mages and Photos Ergna Kavalleratou Dept. of nformaton and Communcaton Systems Engneerng Unversty of the Aegean kavalleratou@aegean.gr Abstract n ths paper, a
More informationKent State University CS 4/ Design and Analysis of Algorithms. Dept. of Math & Computer Science LECT-16. Dynamic Programming
CS 4/560 Desgn and Analyss of Algorthms Kent State Unversty Dept. of Math & Computer Scence LECT-6 Dynamc Programmng 2 Dynamc Programmng Dynamc Programmng, lke the dvde-and-conquer method, solves problems
More informationA Topology-aware Random Walk
A Topology-aware Random Walk Inkwan Yu, Rchard Newman Dept. of CISE, Unversty of Florda, Ganesvlle, Florda, USA Abstract When a graph can be decomposed nto clusters of well connected subgraphs, t s possble
More informationK-means and Hierarchical Clustering
Note to other teachers and users of these sldes. Andrew would be delghted f you found ths source materal useful n gvng your own lectures. Feel free to use these sldes verbatm, or to modfy them to ft your
More informationCracking of the Merkle Hellman Cryptosystem Using Genetic Algorithm
Crackng of the Merkle Hellman Cryptosystem Usng Genetc Algorthm Zurab Kochladze 1 * & Lal Besela 2 1 Ivane Javakhshvl Tbls State Unversty, 1, I.Chavchavadze av 1, 0128, Tbls, Georga 2 Sokhum State Unversty,
More informationCS221: Algorithms and Data Structures. Priority Queues and Heaps. Alan J. Hu (Borrowing slides from Steve Wolfman)
CS: Algorthms and Data Structures Prorty Queues and Heaps Alan J. Hu (Borrowng sldes from Steve Wolfman) Learnng Goals After ths unt, you should be able to: Provde examples of approprate applcatons for
More informationRepeater Insertion for Two-Terminal Nets in Three-Dimensional Integrated Circuits
Repeater Inserton for Two-Termnal Nets n Three-Dmensonal Integrated Crcuts Hu Xu, Vasls F. Pavlds, and Govann De Mchel LSI - EPFL, CH-5, Swtzerland, {hu.xu,vasleos.pavlds,govann.demchel}@epfl.ch Abstract.
More informationVirtual Machine Migration based on Trust Measurement of Computer Node
Appled Mechancs and Materals Onlne: 2014-04-04 ISSN: 1662-7482, Vols. 536-537, pp 678-682 do:10.4028/www.scentfc.net/amm.536-537.678 2014 Trans Tech Publcatons, Swtzerland Vrtual Machne Mgraton based on
More informationCS 534: Computer Vision Model Fitting
CS 534: Computer Vson Model Fttng Sprng 004 Ahmed Elgammal Dept of Computer Scence CS 534 Model Fttng - 1 Outlnes Model fttng s mportant Least-squares fttng Maxmum lkelhood estmaton MAP estmaton Robust
More informationSorting. Sorting. Why Sort? Consistent Ordering
Sortng CSE 6 Data Structures Unt 15 Readng: Sectons.1-. Bubble and Insert sort,.5 Heap sort, Secton..6 Radx sort, Secton.6 Mergesort, Secton. Qucksort, Secton.8 Lower bound Sortng Input an array A of data
More informationType-2 Fuzzy Non-uniform Rational B-spline Model with Type-2 Fuzzy Data
Malaysan Journal of Mathematcal Scences 11(S) Aprl : 35 46 (2017) Specal Issue: The 2nd Internatonal Conference and Workshop on Mathematcal Analyss (ICWOMA 2016) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES
More information3D vector computer graphics
3D vector computer graphcs Paolo Varagnolo: freelance engneer Padova Aprl 2016 Prvate Practce ----------------------------------- 1. Introducton Vector 3D model representaton n computer graphcs requres
More informationCSE 326: Data Structures Quicksort Comparison Sorting Bound
CSE 326: Data Structures Qucksort Comparson Sortng Bound Bran Curless Sprng 2008 Announcements (5/14/08) Homework due at begnnng of class on Frday. Secton tomorrow: Graded homeworks returned More dscusson
More informationQuality Improvement Algorithm for Tetrahedral Mesh Based on Optimal Delaunay Triangulation
Intellgent Informaton Management, 013, 5, 191-195 Publshed Onlne November 013 (http://www.scrp.org/journal/m) http://dx.do.org/10.36/m.013.5601 Qualty Improvement Algorthm for Tetrahedral Mesh Based on
More informationCHAPTER 10: ALGORITHM DESIGN TECHNIQUES
CHAPTER 10: ALGORITHM DESIGN TECHNIQUES So far, we have been concerned wth the effcent mplementaton of algorthms. We have seen that when an algorthm s gven, the actual data structures need not be specfed.
More informationAvailable online at Available online at Advanced in Control Engineering and Information Science
Avalable onlne at wwwscencedrectcom Avalable onlne at wwwscencedrectcom Proceda Proceda Engneerng Engneerng 00 (2011) 15000 000 (2011) 1642 1646 Proceda Engneerng wwwelsevercom/locate/proceda Advanced
More informationComputation of a Minimum Average Distance Tree on Permutation Graphs*
Annals of Pure and Appled Mathematcs Vol, No, 0, 74-85 ISSN: 79-087X (P), 79-0888(onlne) Publshed on 8 December 0 wwwresearchmathscorg Annals of Computaton of a Mnmum Average Dstance Tree on Permutaton
More informationCan We Beat the Prefix Filtering? An Adaptive Framework for Similarity Join and Search
Can We Beat the Prefx Flterng? An Adaptve Framework for Smlarty Jon and Search Jannan Wang Guolang L Janhua Feng Department of Computer Scence and Technology, Tsnghua Natonal Laboratory for Informaton
More information5 The Primal-Dual Method
5 The Prmal-Dual Method Orgnally desgned as a method for solvng lnear programs, where t reduces weghted optmzaton problems to smpler combnatoral ones, the prmal-dual method (PDM) has receved much attenton
More informationInternational Journal of Scientific & Engineering Research, Volume 7, Issue 5, May ISSN Some Polygonal Sum Labeling of Bistar
Internatonal Journal of Scentfc & Engneerng Research Volume 7 Issue 5 May-6 34 Some Polygonal Sum Labelng of Bstar DrKAmuthavall SDneshkumar ABSTRACT- A (p q) graph G s sad to admt a polygonal sum labelng
More informationDistributed Middlebox Placement Based on Potential Game
Int. J. Communcatons, Network and System Scences, 2017, 10, 264-273 http://www.scrp.org/ournal/cns ISSN Onlne: 1913-3723 ISSN Prnt: 1913-3715 Dstrbuted Mddlebox Placement Based on Potental Game Yongwen
More informationVirtual Memory. Background. No. 10. Virtual Memory: concept. Logical Memory Space (review) Demand Paging(1) Virtual Memory
Background EECS. Operatng System Fundamentals No. Vrtual Memory Prof. Hu Jang Department of Electrcal Engneerng and Computer Scence, York Unversty Memory-management methods normally requres the entre process
More informationMathematics 256 a course in differential equations for engineering students
Mathematcs 56 a course n dfferental equatons for engneerng students Chapter 5. More effcent methods of numercal soluton Euler s method s qute neffcent. Because the error s essentally proportonal to the
More informationChapter 6 Programmng the fnte element method Inow turn to the man subject of ths book: The mplementaton of the fnte element algorthm n computer programs. In order to make my dscusson as straghtforward
More informationAn Efficient Algorithm for Minimum Vertex Cover Problem
An Effcent Algorthm for Mnmum Vertex Cover Problem Rong Long Wang Zheng Tang Xn Shun Xu Member Non-member Non-member Paper An effcent parallel algorthm for solvng the mnmum vertex cover problem usng bnary
More informationMachine Learning: Algorithms and Applications
14/05/1 Machne Learnng: Algorthms and Applcatons Florano Zn Free Unversty of Bozen-Bolzano Faculty of Computer Scence Academc Year 011-01 Lecture 10: 14 May 01 Unsupervsed Learnng cont Sldes courtesy of
More informationIntra-Parametric Analysis of a Fuzzy MOLP
Intra-Parametrc Analyss of a Fuzzy MOLP a MIAO-LING WANG a Department of Industral Engneerng and Management a Mnghsn Insttute of Technology and Hsnchu Tawan, ROC b HSIAO-FAN WANG b Insttute of Industral
More informationMULTISPECTRAL IMAGES CLASSIFICATION BASED ON KLT AND ATR AUTOMATIC TARGET RECOGNITION
MULTISPECTRAL IMAGES CLASSIFICATION BASED ON KLT AND ATR AUTOMATIC TARGET RECOGNITION Paulo Quntlano 1 & Antono Santa-Rosa 1 Federal Polce Department, Brasla, Brazl. E-mals: quntlano.pqs@dpf.gov.br and
More informationS1 Note. Basis functions.
S1 Note. Bass functons. Contents Types of bass functons...1 The Fourer bass...2 B-splne bass...3 Power and type I error rates wth dfferent numbers of bass functons...4 Table S1. Smulaton results of type
More informationApproximations for Steiner Trees with Minimum Number of Steiner Points
Journal of Global Optmzaton 18: 17 33, 000. 17 000 Kluwer Academc ublshers. rnted n the Netherlands. Approxmatons for Stener Trees wth Mnmum Number of Stener onts 1, 1,,,,3, DONGHUI CHEN *, DING-ZHU DU
More informationUNIT 2 : INEQUALITIES AND CONVEX SETS
UNT 2 : NEQUALTES AND CONVEX SETS ' Structure 2. ntroducton Objectves, nequaltes and ther Graphs Convex Sets and ther Geometry Noton of Convex Sets Extreme Ponts of Convex Set Hyper Planes and Half Spaces
More informationEVALUATION OF THE PERFORMANCES OF ARTIFICIAL BEE COLONY AND INVASIVE WEED OPTIMIZATION ALGORITHMS ON THE MODIFIED BENCHMARK FUNCTIONS
Academc Research Internatonal ISS-L: 3-9553, ISS: 3-9944 Vol., o. 3, May 0 EVALUATIO OF THE PERFORMACES OF ARTIFICIAL BEE COLOY AD IVASIVE WEED OPTIMIZATIO ALGORITHMS O THE MODIFIED BECHMARK FUCTIOS Dlay
More informationLine Clipping by Convex and Nonconvex Polyhedra in E 3
Lne Clppng by Convex and Nonconvex Polyhedra n E 3 Václav Skala 1 Department of Informatcs and Computer Scence Unversty of West Bohema Unverztní 22, Box 314, 306 14 Plzeò Czech Republc e-mal: skala@kv.zcu.cz
More informationDetermining the Optimal Network Partition and Kanban Allocation in JIT Production Lines
Determnng the Optmal Network Partton and Kanban Allocaton n JIT Producton Lnes Markus Ettl and Markus Schwehm IMMD VII, Unverstät Erlangen-Nürnberg, D-9058 Erlangen, Germany Abstract. One way to reduce
More informationHarvard University CS 101 Fall 2005, Shimon Schocken. Assembler. Elements of Computing Systems 1 Assembler (Ch. 6)
Harvard Unversty CS 101 Fall 2005, Shmon Schocken Assembler Elements of Computng Systems 1 Assembler (Ch. 6) Why care about assemblers? Because Assemblers employ some nfty trcks Assemblers are the frst
More informationAn Application of Network Simplex Method for Minimum Cost Flow Problems
BALKANJM 0 (0) -0 Contents lsts avalable at BALKANJM BALKAN JOURNAL OF MATHEMATICS journal homepage: www.balkanjm.com An Applcaton of Network Smplex Method for Mnmum Cost Flow Problems Ergun EROGLU *a
More informationSimulation Based Analysis of FAST TCP using OMNET++
Smulaton Based Analyss of FAST TCP usng OMNET++ Umar ul Hassan 04030038@lums.edu.pk Md Term Report CS678 Topcs n Internet Research Sprng, 2006 Introducton Internet traffc s doublng roughly every 3 months
More informationSupport Vector Machines
/9/207 MIST.6060 Busness Intellgence and Data Mnng What are Support Vector Machnes? Support Vector Machnes Support Vector Machnes (SVMs) are supervsed learnng technques that analyze data and recognze patterns.
More informationX- Chart Using ANOM Approach
ISSN 1684-8403 Journal of Statstcs Volume 17, 010, pp. 3-3 Abstract X- Chart Usng ANOM Approach Gullapall Chakravarth 1 and Chaluvad Venkateswara Rao Control lmts for ndvdual measurements (X) chart are
More informationLoop Transformations, Dependences, and Parallelization
Loop Transformatons, Dependences, and Parallelzaton Announcements Mdterm s Frday from 3-4:15 n ths room Today Semester long project Data dependence recap Parallelsm and storage tradeoff Scalar expanson
More informationPriority queues and heaps Professors Clark F. Olson and Carol Zander
Prorty queues and eaps Professors Clark F. Olson and Carol Zander Prorty queues A common abstract data type (ADT) n computer scence s te prorty queue. As you mgt expect from te name, eac tem n te prorty
More informationAN IMPROVED GENETIC ALGORITHM FOR RECTANGLES CUTTING & PACKING PROBLEM. Wang Shoukun, Wang Jingchun, Jin Yihui
Copyrght 2002 IFAC 5th Trennal World Congress, Barcelona, Span A IPROVED GEETIC ALGORITH FOR RECTAGLES CUTTIG & PACKIG PROBLE Wang Shouun, Wang Jngchun, Jn Yhu Tsnghua Unversty, Beng 00084, P. R. Chna
More information